Unifying trigonometric and hyperbolic function derivatives via negative integer order polylogarithms

TL;DR

This paper reveals that the polylogarithm function encodes derivatives of both trigonometric and hyperbolic functions at negative integer orders, unifying their derivatives and operator applications into a single framework.

Contribution

It demonstrates that polylogarithms of negative integer order encode derivatives and operator applications of trigonometric and hyperbolic functions, unifying these derivatives in a novel way.

Findings

Polylogarithm functions encode derivatives of trigonometric and hyperbolic functions.

A sum relation connects polylogarithms of order -n to lower orders.

Unified framework for derivatives of special functions at negative integers.

Abstract

Special functions like the polygamma, Hurwitz zeta, and Lerch zeta functions have sporadically been connected with the nth derivatives of trigonometric functions. We show the polylogarithm $\text{Li}_s(z)$, a function of complex argument and order $z$ and $s$, encodes the nth derivatives of the cotangent, tangent, cosecant and secant functions, and their hyperbolic equivalents, at negative integer orders $s = -n$. We then show how at the same orders, the polylogarithm represents the nth application of the operator $x \frac{d}{dx}$ on the inverse trigonometric and hyperbolic functions. Finally, we construct a sum relating two polylogarithms of order $-n$ to a linear combination of polylogarithms of orders $s = 0, -1, -2, ..., -n$.